Evaluate (3-23i)/(16-13i)

$\frac{3 - 23 i}{16 - 13 i}$

Step 1

Multiply the numerator and denominator of $\frac{3 - 23 i}{16 - 13 i}$ by the conjugate of $16 - 13 i$ to make the denominator real.

$\frac{3 - 23 i}{16 - 13 i} \cdot \frac{16 + 13 i}{16 + 13 i}$

Step 2

Multiply.

Combine.

$\frac{(3 - 23 i) (16 + 13 i)}{(16 - 13 i) (16 + 13 i)}$

Simplify the numerator.

Expand $(3 - 23 i) (16 + 13 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{3 (16 + 13 i) - 23 i (16 + 13 i)}{(16 - 13 i) (16 + 13 i)}$

Apply the distributive property.

$\frac{3 \cdot 16 + 3 (13 i) - 23 i (16 + 13 i)}{(16 - 13 i) (16 + 13 i)}$

Apply the distributive property.

$\frac{3 \cdot 16 + 3 (13 i) - 23 i \cdot 16 - 23 i (13 i)}{(16 - 13 i) (16 + 13 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $16$ .

$\frac{48 + 3 (13 i) - 23 i \cdot 16 - 23 i (13 i)}{(16 - 13 i) (16 + 13 i)}$

Multiply $13$ by $3$ .

$\frac{48 + 39 i - 23 i \cdot 16 - 23 i (13 i)}{(16 - 13 i) (16 + 13 i)}$

Multiply $16$ by $- 23$ .

$\frac{48 + 39 i - 368 i - 23 i (13 i)}{(16 - 13 i) (16 + 13 i)}$

Multiply $- 23 i (13 i)$ .

Multiply $13$ by $- 23$ .

$\frac{48 + 39 i - 368 i - 299 i i}{(16 - 13 i) (16 + 13 i)}$

Raise $i$ to the power of $1$ .

$\frac{48 + 39 i - 368 i - 299 (i^{1} i)}{(16 - 13 i) (16 + 13 i)}$

Raise $i$ to the power of $1$ .

$\frac{48 + 39 i - 368 i - 299 (i^{1} i^{1})}{(16 - 13 i) (16 + 13 i)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{48 + 39 i - 368 i - 299 i^{1 + 1}}{(16 - 13 i) (16 + 13 i)}$

Add $1$ and $1$ .

$\frac{48 + 39 i - 368 i - 299 i^{2}}{(16 - 13 i) (16 + 13 i)}$

Rewrite $i^{2}$ as $- 1$ .

$\frac{48 + 39 i - 368 i - 299 \cdot - 1}{(16 - 13 i) (16 + 13 i)}$

Multiply $- 299$ by $- 1$ .

$\frac{48 + 39 i - 368 i + 299}{(16 - 13 i) (16 + 13 i)}$

Add $48$ and $299$ .

$\frac{347 + 39 i - 368 i}{(16 - 13 i) (16 + 13 i)}$

Subtract $368 i$ from $39 i$ .

$\frac{347 - 329 i}{(16 - 13 i) (16 + 13 i)}$

Simplify the denominator.

Expand $(16 - 13 i) (16 + 13 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{347 - 329 i}{16 (16 + 13 i) - 13 i (16 + 13 i)}$

Apply the distributive property.

$\frac{347 - 329 i}{16 \cdot 16 + 16 (13 i) - 13 i (16 + 13 i)}$

Apply the distributive property.

$\frac{347 - 329 i}{16 \cdot 16 + 16 (13 i) - 13 i \cdot 16 - 13 i (13 i)}$

Simplify.

Multiply $16$ by $16$ .

$\frac{347 - 329 i}{256 + 16 (13 i) - 13 i \cdot 16 - 13 i (13 i)}$

Multiply $13$ by $16$ .

$\frac{347 - 329 i}{256 + 208 i - 13 i \cdot 16 - 13 i (13 i)}$

Multiply $16$ by $- 13$ .

$\frac{347 - 329 i}{256 + 208 i - 208 i - 13 i (13 i)}$

Multiply $13$ by $- 13$ .

$\frac{347 - 329 i}{256 + 208 i - 208 i - 169 i i}$

Raise $i$ to the power of $1$ .

$\frac{347 - 329 i}{256 + 208 i - 208 i - 169 (i^{1} i)}$

Raise $i$ to the power of $1$ .

$\frac{347 - 329 i}{256 + 208 i - 208 i - 169 (i^{1} i^{1})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{347 - 329 i}{256 + 208 i - 208 i - 169 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{347 - 329 i}{256 + 208 i - 208 i - 169 i^{2}}$

Subtract $208 i$ from $208 i$ .

$\frac{347 - 329 i}{256 + 0 - 169 i^{2}}$

Add $256$ and $0$ .

$\frac{347 - 329 i}{256 - 169 i^{2}}$

Simplify each term.

Rewrite $i^{2}$ as $- 1$ .

$\frac{347 - 329 i}{256 - 169 \cdot - 1}$

Multiply $- 169$ by $- 1$ .

$\frac{347 - 329 i}{256 + 169}$

Add $256$ and $169$ .

$\frac{347 - 329 i}{425}$

Step 3

Split the fraction $\frac{347 - 329 i}{425}$ into two fractions.

$\frac{347}{425} + \frac{- 329 i}{425}$

Step 4

Move the negative in front of the fraction.

$\frac{347}{425} - \frac{329 i}{425}$

Evaluate (3-23i)/(16-13i)

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